An exchange is like an platform that acts as an intermediary between buyers and sellers of the cryptocurrency. But if you have a desktop wallet like Electrum, then you can used it to send bitcoin to another wallet without going to an exchange. Please go here: https://bitcoin.org/en/choose-your-wallet and choose the best wallet that will fit your needs. I hope I answer your question. LOL.

Use a wallet like Bitcoin Core, Armory or Electrum.
Usually you press Send, enter the address and the amount of BTC and you send it.

That's how you should be doing it, you need to use an exchange to obtain Bitcoin in the first place though.

Bitcoins are not stored on computers, are stored on the Blockchain and you can acces to them with your computer if you own the private keys. Paper wallets are a safest option than exchanges for example.

He is not asking for the safest method to store his coins. Like everyone said,if you have coins you could use a light weight wallet and the receiver must have a wallet address of his choosing,either a web wallet or a light weight wallet or a mobile wallet and start sending your coins,use exchanges or local bitcoins to purchase the coins with your local currency.

I beg to differ with that.  Private keys are not stored on the blockchain but wherever you choose to keep them, so there's little difference between storing a private key and "your bitcoin" on your computer.   Anyway.

I'm not sure what OP means here, but this should probably be in beginners and help section.

If you have an iPhone or an Android smart phone, the EASIEST way would be to download a mobile wallet like Breadwallet and use that to send & transfer BTC anywhere, anytime.

I would also recommend setting one up on your computer as a backup.

You need a third party wallet to generate your keys. Where you store your keys is up to you but in any case you do need a wallet.

This is not true, you can use official bitcoin client or SPV wallets like electrum and jaxx which doesn't store any of your personal details anywhere online. All information of your wallet will be stored locally in your computer or mobile so that there will be no any third party involvement.

Just use wallets I have recommended above to send bitcoin from one person to another without involvement of any one in between. You can neglect involvement of miners and node operators because they can't manipulate your transaction and they are mandatory part of bitcoin ecosystem.

No, that's not the only way. You can use your own wallet and from it you can send Bitcoin to anyone else int he world. No need to use exchanges, really...